Supersaturation of C4: From Zarankiewicz towards Erdős–Simonovits–Sidorenko

Abstract

For a positive integer n, a graph F and a bipartite graph G ⊆ K n,n let F (n + n, G) denote the number of copies of F in G, and let F (n+n, m) denote the minimum number of copies of F in all graphs G ⊆ K n,n with m edges. The study of such a function is the subject of theorems of supersaturated graphs and closely related to the Sidorenko-Erdős-Simonovits conjecture as well. In the present paper we investigate the case when F = K 2,t and in particular the quadrilateral graph case. For F = C 4 , we obtain exact results if m and the corresponding Zarankiewicz number differ by at most n, by a finite geometric construction of almost difference sets. F = K 2,t if m and the corresponding Zarankiewicz number differs by Cn √ n we prove asymptotically sharp results. We also study stability questions and point out the connections to covering and packing block designs.

The Turán number of blow-ups of trees

“…Note that n 2− 1 r is the order of magnitude of the Turán function of K r,t . In special cases, the supersaturation is even more understood when the edge cardinality is in the interval [ex(n, K r,t ), (1 + ε) ex(n, K r,t )], see the paper of the third author [15] for exact results in the case r = t = 2 and on the dependence of c r,t (γ) on γ.…”

Unavoidable patterns in locally balanced colourings

“…Suppose that the blue density between V i and V j is at least δ. It follows that the blue subgraph of G[V i ∪V j ] must contain at least 2 −10 δ 4 (n/C) 4 copies of a cycle on 4 vertices (see, for example, Theorem 1.9(iv) from [14]). At most 2δ 5 (n/C) 4 such 4-cycles can contain a blue edge from G[V i ] or G[V j ], by the density assumption.…”

Regular Turán numbers of complete bipartite graphs

“…While Füredi's constructions are algebraically defined graphs, ours will be written as Cayley sum graphs. We note that in [29] and [30] bipartite K 2,t -free graphs of the same flavor are constructed.…”